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Identifier

Mp00200: Binary words —twist⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000277: Integer compositions ⟶ ℤ

Values

0 => 1 => [1] => 1

1 => 0 => [1] => 1

00 => 10 => [1,1] => 1

01 => 11 => [2] => 1

10 => 00 => [2] => 1

11 => 01 => [1,1] => 1

000 => 100 => [1,2] => 2

001 => 101 => [1,1,1] => 1

010 => 110 => [2,1] => 2

011 => 111 => [3] => 1

100 => 000 => [3] => 1

101 => 001 => [2,1] => 2

110 => 010 => [1,1,1] => 1

111 => 011 => [1,2] => 2

0000 => 1000 => [1,3] => 3

0001 => 1001 => [1,2,1] => 5

0010 => 1010 => [1,1,1,1] => 1

0011 => 1011 => [1,1,2] => 3

0100 => 1100 => [2,2] => 5

0101 => 1101 => [2,1,1] => 3

0110 => 1110 => [3,1] => 3

0111 => 1111 => [4] => 1

1000 => 0000 => [4] => 1

1001 => 0001 => [3,1] => 3

1010 => 0010 => [2,1,1] => 3

1011 => 0011 => [2,2] => 5

1100 => 0100 => [1,1,2] => 3

1101 => 0101 => [1,1,1,1] => 1

1110 => 0110 => [1,2,1] => 5

1111 => 0111 => [1,3] => 3

00000 => 10000 => [1,4] => 4

00001 => 10001 => [1,3,1] => 11

00010 => 10010 => [1,2,1,1] => 9

00011 => 10011 => [1,2,2] => 16

00100 => 10100 => [1,1,1,2] => 4

00101 => 10101 => [1,1,1,1,1] => 1

00110 => 10110 => [1,1,2,1] => 9

00111 => 10111 => [1,1,3] => 6

01000 => 11000 => [2,3] => 9

01001 => 11001 => [2,2,1] => 16

01010 => 11010 => [2,1,1,1] => 4

01011 => 11011 => [2,1,2] => 11

01100 => 11100 => [3,2] => 9

01101 => 11101 => [3,1,1] => 6

01110 => 11110 => [4,1] => 4

01111 => 11111 => [5] => 1

10000 => 00000 => [5] => 1

10001 => 00001 => [4,1] => 4

10010 => 00010 => [3,1,1] => 6

10011 => 00011 => [3,2] => 9

10100 => 00100 => [2,1,2] => 11

10101 => 00101 => [2,1,1,1] => 4

10110 => 00110 => [2,2,1] => 16

10111 => 00111 => [2,3] => 9

11000 => 01000 => [1,1,3] => 6

11001 => 01001 => [1,1,2,1] => 9

11010 => 01010 => [1,1,1,1,1] => 1

11011 => 01011 => [1,1,1,2] => 4

11100 => 01100 => [1,2,2] => 16

11101 => 01101 => [1,2,1,1] => 9

11110 => 01110 => [1,3,1] => 11

11111 => 01111 => [1,4] => 4

000000 => 100000 => [1,5] => 5

000001 => 100001 => [1,4,1] => 19

000010 => 100010 => [1,3,1,1] => 26

000011 => 100011 => [1,3,2] => 40

000100 => 100100 => [1,2,1,2] => 40

000101 => 100101 => [1,2,1,1,1] => 14

000110 => 100110 => [1,2,2,1] => 61

000111 => 100111 => [1,2,3] => 35

001000 => 101000 => [1,1,1,3] => 10

001001 => 101001 => [1,1,1,2,1] => 14

001010 => 101010 => [1,1,1,1,1,1] => 1

001011 => 101011 => [1,1,1,1,2] => 5

001100 => 101100 => [1,1,2,2] => 35

001101 => 101101 => [1,1,2,1,1] => 19

001110 => 101110 => [1,1,3,1] => 26

001111 => 101111 => [1,1,4] => 10

010000 => 110000 => [2,4] => 14

010001 => 110001 => [2,3,1] => 40

010010 => 110010 => [2,2,1,1] => 35

010011 => 110011 => [2,2,2] => 61

010100 => 110100 => [2,1,1,2] => 19

010101 => 110101 => [2,1,1,1,1] => 5

010110 => 110110 => [2,1,2,1] => 40

010111 => 110111 => [2,1,3] => 26

011000 => 111000 => [3,3] => 19

011001 => 111001 => [3,2,1] => 35

011010 => 111010 => [3,1,1,1] => 10

011011 => 111011 => [3,1,2] => 26

011100 => 111100 => [4,2] => 14

011101 => 111101 => [4,1,1] => 10

011110 => 111110 => [5,1] => 5

011111 => 111111 => [6] => 1

100000 => 000000 => [6] => 1

100001 => 000001 => [5,1] => 5

100010 => 000010 => [4,1,1] => 10

100011 => 000011 => [4,2] => 14

100100 => 000100 => [3,1,2] => 26

100101 => 000101 => [3,1,1,1] => 10

100110 => 000110 => [3,2,1] => 35

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Description

The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' Mp00071descent composition from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals
$$ \sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell}, $$
where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.

Map

twist

Description

Return the binary word with first letter inverted.